In the last decades, the polymerization process has become an important topic in both applied and basic research [1–5]. It is commonly believed that in dilute solutions [6], where interactions between different polymers can be neglected, linear polymers can be modeled by selfavoiding walks (SAWs) [1,2,5] and branched polymers are in the universality class of lattice animals (LAs) [7–9]. The SAW ensemble consists of all configurations of nointersecting random walks (of N steps), while the LA ensemble consists of all configurations of N-site clusters. In recent years it became clear that asymptotically SAWs can be generated by a kinetic growth walk (KGW), where at each step the random walker can move only to those neighboring sites that have not been visited before [10– 12]. Very recently, in order to generate branched polymer structure, Lucena et al. [13] generalized the KGW to include branching [branched polymer growth model (BPGM)] [14]. They found the interesting phenomenon that at a (small) finite probability of branching, bc, that increases monotonically with the concentration q of impurities in the system, a transition occurs from SAW-type structures at small branching probability b to compact structures at large b. The important question, however, to which universality class the structures at the critical line bc q belong to has not been resolved. In this Letter we show that the branched polymers at the critical line are not of LA type as would have been expected, but most probably belong to the universality class of percolation. Our results are based on extensive numerical studies of the fractal dimension dmin of the minimum path between two cluster points, of the chemical dimension d and of the exponent t characterizing the distribution of clusters of given size. The BPGM generates polymer structures from a seed in a self-avoiding manner similar to the KGW, but allows for the possibility of branching with bifurcation probability b. To be specific, consider a square lattice where at t 0 the center of the lattice is occupied by a polymer “seed.” There are four empty nearest-neighbor sites of the seed, where the polymer is allowed to grow. At step t 1, two of these four growth sites are chosen randomly: One of them is occupied by the polymer with probability 1, the other is occupied with probability b. This process is continued. At step t 1 1, the polymer can grow from each of the sites added at the foregoing step t to empty nearest-neighbor sites (growth sites) either in a linear fashion or by bifurcation with probability b, provided there are enough growth sites left; otherwise, the polymer stops growing. If a certain concentration q of the lattice sites are occupied by impurities and cannot serve as growth sites, large polymers can be generated only below the percolation threshold qc of the considered lattice qc 0.407 23 on the square lattice). According to Lucena et al. [13] the critical line bc q separates a phase that belongs to the universality class of SAWs at small b from a phase belonging to the universality class of compact Eden clusters at large b, with bc 0 0.055 and bc qc 1. The universality class at the critical line could not be identified by Lucena et al. [13] when calculating the fractal dimension df of the clusters in Euclidean space. To determine the universality class of the BPGM at the critical line we have studied the growth process in chemical space and determined the critical exponents dmin, d , and t [14]. The chemical distance between two cluster points separated by Euclidean distance r is defined as the length of the shortest path between them on the cluster, and the fractal dimension dmin describes how scales with r, rdmin . The chemical dimension d describes how the cluster mass within chemical distance scales with , M d . The conventional fractal dimension df , defined by M rdf , is related to d and dmin by df d dmin. For SAW structures, d 1 and dmin df , while for Eden clusters dmin 1 and d df d. For percolation clusters in d 2 at criticality, we have dmin 1.13, df 91 48, and d 1.68 [15]. The exponent t describes the cluster size distribution. We have found that direct numerical calculations of df by the mass-radius relation are not conclusive due to the existence of strong boundary effects. In contrast, calculations in chemical space do not have any boundary effect, since the branched polymers are grown in chemical space: At step t 1 the first shell in space is completed, at t 2 the second shell, and so on. Hence the chemical space is the natural metric for calculating the critical exponents of the BPGM.